2. IJAMSS ESTIMATION OF POPULATIO Fixed 00 New 1

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by  IASET JOURNALS

International Journal of Applied Mathematics
& Statistical Sciences (IJAMSS)
ISSN(P): 2319-3972; ISSN(E): 2319-3980
Vol. 6, Issue 1, Dec – Jan 2017; 19-32
© IASET

ESTIMATION OF POPULATION PROPORTION USING TWO PHASE SAMPLING
SCHEME IN THE PRESENCE OF NON-RESPONSE
B. B. KHARE1, P. S. JHA2 & S. KHARE3
1,3Department of Statistics, Banaras Hindu University, Varanasi, India
2Department of Molecular & Human Genetics, Banaras Hindu University, Varanasi, India

ABSTRACT
In this paper we have proposed two classes of two phase sampling estimators for population proportion using
auxiliary character in the presence of non-response on the study character. Some members of the proposed classes of
estimators are given. The expressions for bias and mean square error of the proposed classes of estimators are obtained in
case of fixed sample size. Also we obtained the optimum value of sample size and sampling fraction for fixed cost and
fixed precision. An empirical study has been made with the support of the proposed class of estimators.
KEYWORDS: Non-Response, Mean Square Error, Relative Bias, Auxiliary Characters, Attribute, Two Phase Sampling
1. INTRODUCTION
Sometimes, it may not be possible to collect the complete information for all the units selected in the sample due
to lack of interest, person not present at home, lack of knowledge regarding the survey, ethical problems and due to refusal
of the respondent to respond for the given questionnaire. In these situations, Hansen and Hurwitz (1946) have suggested a
technique of sub sampling units from non-respondents. Using Hansen and Hurwitz (1946) technique, Cochran (1977) and
Rao (1986, 1990) have proposed the estimators for population mean in case of known population mean of auxiliary
character.
In sample survey, the use of auxiliary character has a significant role in the estimation of population parameters
such as population mean, ratio and product of two population means, coefficient of variation etc. Several research works
have been done for the estimation of population parameter using auxiliary character in the presence of non-response when
the population mean of the auxiliary character is known / unknown. Further several research works on the estimation of
population mean have been done by Khare and Srivastava (1993, 1995, 2000, 2010) and Singh and Kumar (2009, 2010).
Sometimes auxiliary information is available in the form of auxiliary attribute which may also be used in the estimation of
population mean of the study character. With the help of auxiliary attribute, several authors have proposed different type of
estimators for the estimation of population mean of the study character. In this situation Naik and Gupta (1996), Jhajj et al.
(2006), Shabbir and Gupta (2007, 2010), Singh et.al (2008), Koyuncu (2012), Abd-Elfattah et al. (2010), Singh and
Solanki (2012), Malik and Singh (2013) and Sinha and Kumar (2013, 2014) have proposed different type of estimators for
population mean using the information on auxiliary character as a qualitative character.
In sample surveys it generally happens that the study variable is also available in the form of qualitative characteristic such
as smoking habit, educational status, socio economic group etc. For example if we want to estimates the proportion of
woman smoking in a city, we may take their educational status, socio economic group and monthly expenditure etc as

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20 B. B. Khare, P. S. Jha & S. Khare
auxiliary character. In this situation Singh et al. (2010) have suggested a family of estimators for the population proportion.
Further, Khare et al. (2015) have proposed two classes of estimators for population proportion P using auxiliary character
x in the presence of non-response which are given as follow:
T = H(w,v ) and T = H(w,v ) (1.1)
c1 1 c2 2
*
x x
Such that H (P, 1)= P andH (P, 1)= P, wherew= p*,v = , v = and
i 1i 1 X 2 X
 ¶ H(w,v )
H (P,1) =  i  , " i =1, 2 . Here p* and x* denote the estimators for P and X using Hansen
1i  ¶ w 
(P,1)
and Hurwitz (1946) technique and x and X denote the sample mean based on n units and population mean of the
auxiliary characterx.
In this paper, we have proposed two classes of two phase sampling estimators for population proportion using
auxiliary character in the presence of non-response on the study character. The expressions for the bias and mean square
error of the proposed classes of estimators are obtained and the properties of the proposed classes of estimators are studied
for fixed sample sizes (n¢,n), for a fixed cost (C £ C ) and also for a specified precision (V =V ). Some members of
0 0
the proposed classes of estimators are also obtained and their properties are studied. An empirical study is also given in the
support of proposed classes of the estimators forP.
2. THE ESTIMATORS
Let f and x denote the value of study attribute and auxiliary character for ith unit of the population (i=1, 2, ..., ...,
i i
f
N). In this case will take value 1 if it possessing the attribute otherwise zero. The population is supposed to be divided
i
into N responding and N non-responding units such thatN +N = N .
1 2 1 2
Let P = N∑f /N be the proportion of the units in the population possessing the attributef , P = N∑1f /N be
i 1 i 1
i=1 i=1
the proportion of the units in the responding part of the population possessing the attribute f and P = N∑2f /N is the
2 i 2
i=1
f
proportion of the units in the non responding part of the population possessing the attribute .
When population mean (X) of auxiliary character x is unknown, then it is suggested to select a first phase
sample of size n¢(< N) from the population of size N by using SRSWOR method of sampling and estimate the
¢ ¢ ¢
population mean X by using x , where x denotes the sample mean based on n observations. Again a second phase
sample of size n(< n¢) is selected from the first phase sample of size n¢ by using SRSWOR method of sampling. We
observed that for the study attributef , only n units are responding and n units are not responding in the sample of size
1 2
n. Again we draw a subsample of sizer(= n /k,k >1) from n non-responding units and collect information on r
2 2

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Estimation of Population Proportion Using Two Phase Sampling Scheme in 21
the Presence of Non-Response
units on the study character using personal interview method and applying extra efforts. Now using Hansen and Hurwitz
(1946) technique, we propose the estimator for population proportion Pwhich is given as follows:
n n
p* = 1 p + 2 p¢ , (2.1)
1 2
n n
where p = n∑1f /n is the proportion of the units possessing the attribute f for n responding units in the
1 i 1 1
i=1
r
sample of size n and p¢ = ∑f /r is the proportion of the units possessing the attribute f for r sub-sampled units
2 i
i=1
from n non respondents.
2
*
The variance of p is given as follows:
f W (k - 1)
V(p*) = Sf2 + 2 Sf2(2), (2.2)
n n
N - n
WhereW2 = N2 /N , f =1- n/ N , Q =1- P, Q2 =1- P2, Sf2 = n(n- 1)PQ and
N - n
Sf2(2) = n (2n - 21) P2Q2 are the population mean square of the attribute f for the whole population and for the non-
2 2
responding part of the population.
Similarly, for the corresponding n +r units related to the study attributef , the estimator x* for the population
1
mean X is given by
n n
x* = 1 x + 2 x¢ , (2.3)
1 2
n n
Where x and x¢ are the sample means of the auxiliary character based on n responding units and r sub
1 2 1
sample units from n non responding units.
2
*
The variance of x is given as follows:
f W (k - 1)
V(x*) = S2 + 2 S2 , (2.4)
x x(2)
n n
Where S2and S2 are the population mean squares of auxiliary character x for the whole population and for
x x(2)
the non-responding part of the population.
The estimators for the population mean X based on sample size n¢ and n are given by

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22 B. B. Khare, P. S. Jha & S. Khare
1 n¢ 1 n
x¢ = ∑x And x = ∑x (2.5)
¢ i i
n i=1 ni=1
And their variances are given by
¢
f f
V(x¢) = S2 And V(x) = S2 (2.6)
¢ x x
n n
Where f ¢ =1- n¢ /N
In case when population mean X of auxiliary character x is unknown then we estimate it by x¢ . Further in case
of non-response in the sample of size n drown from n¢ , two situations arises
• When incomplete information on study attribute f and the information on the corresponding n + r units of
1
auxiliary character x is used from the sub-sample of size n selected from n¢ first phase sample. Now we
propose the class of estimators T for population proportion P using auxiliary character x which is given as
u1
follows:
T = H(w,u ), such that H(P, 1)= P and H (P, 1)=1, (2.7)
u1 1 1
x*  ¶ H(w,u )
where w= p*, u = and H (P,1) =  1  .
1 x¢ 1  ¶ w 
(P,1)
• When incomplete information on n + r units of the study attributes f and the complete information on
1
¢
auxiliary character x from the selected sample of size n from n is used and we propose the class of estimators
T for the population proportion P using auxiliary character x which is given as follows:
u2
T = H(w,u ), such that H(P, 1)= P and H (P, 1)=1, (2.8)
u2 2 1
x  ¶ H(w,u )
Whereu = , and H (P,1) =  2  .
2 x¢ 1  ¶ w 
(P,1)
The function H(w, ui)i =1, 2 also satisfies the following regularity conditions:
• Whenever be the sample chosen, w and ui assume positive values in a bounded closed convex subset Di
of the two dimensional real space containing the point (P,1).
• The first and second partial derivatives of H(w, ui) with respect to w and ui exit and are assumed to be
continuous and bounded in two dimensional real spaces D . (2.9)
i
Now, expanding H(w, u ) about the point (Y, 1) = M by using Taylor’s series up to the second order
i
derivatives, we have

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Estimation of Population Proportion Using Two Phase Sampling Scheme in 23
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T = H(P, 1) +(w- P) H (M) + (u - 1)H (M)
ui 1 i 2(i)
{ }
1
+ (w- P)2H (m*)+(u - 1)2H (m*)+2(w- P)(u - 1)H (m*) (2.10)
11 i 22(i) i 12(i)
2

After puttingH(P, 1)= P, H (M)=1 and H (M)=0, we have
1 11
1
T = w+(u - 1)H (M) + {(u - 1)2H (m*)+2(w- P)(u - 1)H (m* )} (2.11)
ui i 2(i) i 22(i) i 12(i)
2
Where
¶ H(m) ¶ H(m) ¶ 2 H(m* )
H1(M) =  ¶ w  , H2(i)(M) =  ¶ u  , H11 (m* )= ¶ w2 ,
M i M
¶ 2 H(m* ) ¶ 2 H(m*)
H (m*)= , H (m*) = , m=(w,u ), m* =(w*,u*), w* =P+θ (w- P),
22(i) ¶ u2 12(i) ¶ w¶ u i i i
i i
u* =1+θ (u - 1), 0<q <1.
i i i i
3. BIAS AND MEAN SQUARE ERROR (MSE) OF THE PROPOSED CLASSES OF ESTIMATORS
Under the regularity conditions specified for the functionH(w, u ), bias and mean square error of the estimator
i
T always exists. In order to derive the expressions for bias and mean square error of the estimators, we use the large
ui
sample approximations.
Let p* =P(1+ε0), x* = X(1+ε1), x = X(1+ε2), x¢ = X(1+ε3), such that E(εℓ)= 0 and εℓ <1
" ℓ
=0, 1, 2, 3.
Now, using SRSWOR method of sampling, we have
f W (k- 1) f W (k - 1) f
E(e02) = Cf2+ 2 Cf2(2), E(ε12)= Cx2 + 2 Cx2(2), E(ε22 ) = Cx2,
n n n n n

f¢ f W (k- 1) f f ¢
E(ε32)= ¢ Cx2, E(ε0ε1)= Cfx + 2 Cfx(2), E(ε0ε2 )= Cfx, E(ε0ε3)= ¢ Cfx ,
n n n n n
where Cf2 = PSf22 , Cf2(2) = SPf2(22) , Cx2 = XSx22 , Cx2(2) = SXx2(22) , Cfx = rfx SPfXSx ,

S S
Cfx(2) = rfx(2) f(2) x(2) and rfx and rfx(2) arethe point-by serial correlation coefficient between f
PX

and x for the whole population and for the non-responding part of the population.

ε ε ε
Now retaining the terms of , and , upto second order only in the expression of bias and mean square
0 1 2

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24 B. B. Khare, P. S. Jha & S. Khare
- 1
error, the expressions for bias and mean square error of the estimator T up to terms of order(n )are given as follows:
ui
11 1  W (k - 1) 
Bias(T ) =  - C2 + 2 C2 H (M)
u1 ¢ x x(2) 22(1)
2n n  n 
, (3.1)
1 1  W (k - 1)  
+2P - ¢ Cfx + 2 Cfx(2)H12(1)(M)
n n  n  
1 1 1   1 1   
Bias(Tu2) =  - ¢ Cx2H22(1)(M)+2P - ¢ CfxH12(1)(M), (3.2)
2n n   n n   
1 1 W (k- 1) 
MSE(T )=V(p*)+ - C2+ 2 C2 H2 (M)
u1 ¢ x x(2) 2(1)
n n  n 
, (3.3)
1 1 W (k- 1) 
+ 2P - ¢ Cfx+ 2 Cfx(2)H2(1)(M)
n n  n 
and
[ ]
1 1 
MSE(Tu2) =V(p*)+ - ¢ H22(2)(M)Cx2 +2PH2(2)(M)Cfx . (3.4)
n n 
The optimum values of H (M)and H (M)which minimize the MSE of the estimatorT , i =1, 2 are
2(1) 2(2) ui
given as follows:
1 1  W (k - 1)
 - C + 2 C
¢ fx fx(2)
n n  n
H (M) = - P (3.5)
2(1) opt 1 1  W (k - 1)
 - C2 + 2 C2
¢ x x(2)
n n  n
and
C
H (M) = - P fx . (3.6)
2(2) opt C2
x
Now putting the optimum value of H (M) and H (M) in the expression of minimum value of
2(1) opt 2(2) opt
MSE(T ) and MSE(T ) given by (3.3) and (3.4) respectively, we get
u1 u2
1 1  W (k - 1) 2
 - ¢ Cfx + 2 Cfx(2)
n n  n 
MSE(T ) =V(p*)- P2 (3.7)
u1 min 1 1  W (k - 1)
 - C2 + 2 C2
¢ x x(2)
n n  n
and

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Estimation of Population Proportion Using Two Phase Sampling Scheme in 25
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2
MSE(T ) =V(p*)- P21 - 1 (Cfx) (3.8)
u2 min n n¢  C2
x
The MSE(T ) given by (3.7) may be approximated to
u1 min
MSE(Tu1)min =V(p*)- P2 CCf2x 1n - n1¢ Cfx +W2(nk - 1)Cfx(2), for CCf2x = CCf2x(2) (3.9)
x x x(2)
Also, the expression of MSE(T ) and MSE(T ) from equation (3.9) and (3.8) is expressed in the
u1 min u2 min
1 1
form of the coefficient of and which are given by
¢
n n
MSE(Tu1)min=1nP2{(Cf2+W2(k- 1)Cf22)- (CCfx2)2 - W2(k- 1))(CfCxC2fx2)}+n1¢ P2 (CCfx2)2 - N1 Cf2 (3.10)
x x x
2 2
MSE(Tu2)min =1nP2{(Cf2 +W2(k- 1)Cf22)- (CCfx2) )}+ n1¢ P2 (CCfx2) - N1 Cf2 (3.11)
x x
The optimum value of H (M) and H (M) in equation (3.5) and (3.6) are in the form of the
2(1) opt 2(2) opt
value of unknown parameters. Sometimes the unknown constant used in the estimators are in the form of some unknown
parameters. In these situations, the optimum value of the constant may be obtained from the past data (Reddy 1978) and if
the information on these parameters are not available from past data, then one may estimate it on the basis of sample
observations without having any loss in efficiency of the proposed estimators. It has been shown that up to the terms of
- 1
order (n ), the minimum values of the mean square error of the estimator is unchanged if we estimate the optimum
values of constants by using the sample values (Srivastava and Jhajj 1983). Any parametric function H(w, u ) satisfying
i
the regularity condition (2.7) can generate a class of asymptotic estimators. Some members of the proposed class of
estimators are given as follows:
T = w(a+(1- a)u ), T = w(ua ),
i1 i i2 i
T = (w+a (1- u ))ub , T = w(2- ug ),
i3 1 i i i4 i
 u - 1 u - 1
Ti5 = w1+a 0uii +1 and Ti6 = w exp(uii +1) , i =1,2 (3.12)
Where a, a , a , a , b and g are constants
1

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26 B. B. Khare, P. S. Jha & S. Khare
4. DETERMINATION OFn¢ , n AND k FOR A FIXED COST (C £ C0)
Let C be the total cost (fixed) of the survey apart from the overhead cost. The cost function C¢ for the cost
0
incurred in the survey apart from overhead expenses, can be expressed by
n
C¢ =C¢ n¢ +C n+C n +C 2 (4.1)
1 1 2 1 3
k
Since C¢ will vary from sample to sample, so we consider the expected cost C to be incurred in the survey apart
from overhead expenses, which is given by
 W 
C = E(C¢) =C¢1n¢ +nC1 +C2W1 +C3 2, (4.2)
 k 
Where
¢
C : The cost per unit of identifying and observing auxiliary variable,
1
C : The cost per unit of mailing questionnaire/visiting the unit at the second phase,
1
C : The cost per unit of collecting and processing data for the study variable f for responding units and
2
C : The cost per unit of obtaining and processing data for the study variable f (after extra efforts) from sub-
3
sampled units from n non-responding units.
2
( )
From equation (3.9) and (3.8) the MSE T ,i =1,2; can be expressed in terms of the notationY , Y
ui min 0i 1i
Y
and which is given as
2i
1 1 k
MSE(T ) = Y + Y + Y +terms of independent of n¢ and n, (4.3)
ui ¢ 1i 0i 2i
n n n
1 1 k
Y = Y = Y =
Where coefficient of terms, coefficient of terms and coefficient of terms.
1i ¢ 0i 2i
n n n
Now we minimize MSE for the fixed cost C £ C and to obtain the optimum values of n¢ , n and k.
0
Let us consider a function
  W  
W = MSE(T )+l C¢ n¢ +nC +C W +C 2 - C , i =1,2 (4.4)
ui i 1 1 2 1 3 0
  k  
l
Where are Lagrange’s multipliers
i
Now differentiating W with respect to n¢ , n and k and equating to zero, we get

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Estimation of Population Proportion Using Two Phase Sampling Scheme in 27
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Y
n¢ = 1i ,(4.5)
opt l C¢
i 1
Y +k Y
n = 0i opt 2i , (4.6)
opt
 
W
l C +C W +C 2 
i 1 2 1 3 
k
 opt 
Y
n
= 2i
And (4.7)
k l C W
i 3 2
Solving (4.6) and (4.7), we have
Y C W
k = 0i 3 2 (4.8)
opt. Y (C +C W )
2i 1 2 1
Putting the values ofn¢ , n and k in (4.4), we have
 
1 W
l =  Y C¢ + (Y +k Y )(C +C W +C 2 ) (4.9)
i 1i 1 0i opt 2i 1 2 1 3
C  k 
0  opt. 
( )
The minimum value of MSE T ,i =1,2is given by
i
2
MSE(T ) = 1  Y C¢ + (Y +k Y )(C +C W +C W2 ) - Sf2 (4.10)
ui min. 1i 1 0i opt. 2i 1 2 1 3
C  k  N
0  opt. 
*
In case of P , the expected total cost is given by
 W 
C = E(C¢)= nC +C W +C 2  (4.11)
1 2 1 3
 k 
1  W 2 Sf2
and MSE(P*)min. = C0  (Sf2 +W2(k - 1)Sf22)(C1 +C2W1 +C3 k2) - N (4.12)
5. DETERMINATION OFn¢ , n AND k FOR A SPECIFIED VARIANCE V =V¢
0
Let V¢ be the variance of the estimator MSE(T ) ,i =1,2 fixed in advance and we have
0 ui min
2
1 1 k Sf
V¢ = Y + Y + Y - (5.1)
0 0i ¢ 1i 2i
n n n N
( )
For minimizing the average total cost Cfor the specified variance of the Estimator MSE T ,i =1,2 , we
ui min

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28 B. B. Khare, P. S. Jha & S. Khare
define a function
  W 
W = C¢ n¢ +nC +C W +C 2 +m {MSE(T )- V¢ }, (5.2)
1 1 1 2 1 3 i ui 0
  k 
where m (i =1,2) are Lagrange multiplier.
i
Now differentiating with respect to n¢ , n and k and equating to zero, we have
m Y
n¢ = i 1i , (5.3)
opt ¢
C
1
m (Y +k Y )
n = i 0i opt 2i , (5.4)
opt
 
W
C +C W +C 2 
 1 2 1 3 
k
 opt. 
m Y
n
= i 2i
and (5.5)
k C W
3 2
From (5.2) , (5.4) and (5.5), we have
Y C W
k = 0i 3 2 , (5.6)
opt. Y (C +C W )
2i 1 2 1
 
 
1 W
and m =  C¢ Y + (Y +k Y )C +C W +C 2  (5.7)
i V¢ + Sf2  1 1i 0i opt. 2i  1 2 1 3 kopt. 
0
 N 
 
The minimum expected total cost for the specified variance V¢ for the estimator MSE(T ) ,i =1,2is given
0 ui min
by
2
 
 
W
 C¢ Y + (Y +k Y )C +C W +C 2 
 1 1i 0i opt. 2i  1 2 1 3 k 
  opt. 
C(T ) = , i =1,2 (5.8)
ui min.
 2
S
V¢ + f 
0
 N 
 
*
In case of P , the fixed precision is given by
2
1 k Sf
V¢ = Y + Y - (5.9)
0 0 1
n n N

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